Definition

Definition

equipped with a further left AA-representationA→ℬ(N)A \to \mathcal{B}(N) by adjointable operators, hence such that ⟨a−,−⟩=⟨−,a*−⟩\langle a -,- \rangle = \langle -,a^\ast -\rangle for all a∈Aa \in A.

A isomorphism between two (A,B)(A,B)-bimodules (N1,⟨−,−⟩)→(N2,⟨−,−⟩2)(N_1, \langle -,-\rangle) \to (N_2, \langle -,-\rangle_2) is a linear operatorN1→N2N_1\to N_2 which is unitary with respect to ⟨−,−⟩2\langle -,-\rangle_2.

Definition

Given an (A,B)(A,B)-Hilbert bimodule (N1,⟨−,−⟩1)(N_1, \langle -,-\rangle_1) and a (B,C)(B,C)-Hilbert bimodule (N2,⟨−,−⟩2)(N_2, \langle -,-\rangle_2), the tensor product of Hilbert bimodulesN1⊗BN2N_1 \otimes_B N_2 is the (A,C)(A,C)-Hilbert bimodule obtained from the ordinary (algebraic) tensor product of modules over ℂ\mathbb{C}N1⊗ℂN2N_1 \otimes_{\mathbb{C}} N_2 by